The sum `2xx5+5xx9+8xx13+…10` terms is

Options

A. `4500`
B. `4555`
C. `5454`
D. None of these

Explanation

To find the sum of the series \(2 \times 5 + 5 \times 9 + 8 \times 13 + \ldots\) for 10 terms, we will first identify the patterns in the series and then derive a formula for the sum.

### Step 1: Identify the sequences
The terms in the series can be expressed as products of two sequences:
- The first sequence: \(2, 5, 8, \ldots\)
- The second sequence: \(5, 9, 13, \ldots\)

### Step 2: Find the general term for each sequence
1. **First Sequence**: This is an arithmetic sequence where the first term \(a_1 = 2\) and the common difference \(d_1 = 3\).
- The \(n\)-th term of the first sequence can be expressed as:
\[ a_n = 2 + (n - 1) \cdot 3 = 3n - 1 \]

2. **Second Sequence**: This is also an arithmetic sequence where the first term \(b_1 = 5\) and the common difference \(d_2 = 4\).
- The \(n\)-th term of the second sequence can be expressed as:
\[ b_n = 5 + (n - 1) \cdot 4 = 4n + 1 \]

### Step 3: Write the general term of the series
The \(n\)-th term of the series can be expressed as:
\[ T_n = a_n \cdot b_n = (3n - 1)(4n + 1) \]

### Step 4: Expand the general term
Now, we expand \(T_n\):
\[ T_n = (3n - 1)(4n + 1) = 12n^2 + 3n - 4n - 1 = 12n^2 - n - 1 \]

### Step 5: Find the sum of the first 10 terms
We need to find the sum \(S_{10} = \sum_{n=1}^{10} T_n\):
\[ S_{10} = \sum_{n=1}^{10} (12n^2 - n - 1) \]
This can be split into three separate summations:
\[ S_{10} = 12 \sum_{n=1}^{10} n^2 - \sum_{n=1}^{10} n - \sum_{n=1}^{10} 1 \]

### Step 6: Use formulas for summation
We will use the following formulas:
1. \(\sum_{n=1}^{N} n^2 = \frac{N(N + 1)(2N + 1)}{6}\)
2. \(\sum_{n=1}^{N} n = \frac{N(N + 1)}{2}\)
3. \(\sum_{n=1}^{N} 1 = N\)

For \(N = 10\):
- \(\sum_{n=1}^{10} n^2 = \frac{10 \cdot 11 \cdot 21}{6} = 385\)
- \(\sum_{n=1}^{10} n = \frac{10 \cdot 11}{2} = 55\)
- \(\sum_{n=1}^{10} 1 = 10\)

### Step 7: Substitute back into the sum
Now substituting these values back into the equation for \(S_{10}\):
\[ S_{10} = 12 \cdot 385 - 55 - 10 \]
\[ S_{10} = 4620 - 55 - 10 = 4555 \]

### Final Result
The sum of the series for the first 10 terms is:
\[ \boxed{4555} \]